Problems with worked out solutions about acceleration of electrons in the electric field of an Electron Gun

Problem 1)

Explain the influence of the heater voltage and the influence of the acceleration voltage on the electron beam of an Electron Gun.

Problem 2)

The acceleration voltage of an Electron Gun is Va = 500 V and the distance between Kathode and Anode is d = 5cm. Non-relativistic calculation can be used.
  1. Calculate how fast the electrons are moving when they pass the anode.
  2. Find the work done by the electric field on an electron.
  3. What time does the electron take to get from the hot cathode to the anode?
  4. Now the acceleration voltage changes and electrons are passing the anode with a speed of $v=2{,}054\cdot 10^7 \,\frac{\text m}{\text s}$. Find the acceleration voltage.

Problem 3)

The acceleration voltage of an electron gun is Va = 8 kV.
  1. Is non-relativistic calculation correct or must the priciples of relativity benn taken into account? Give reasons for your answer.
  2. Calculate how fast the electrons are moving when they pass the anode.
  3. Now the acceleration voltage changes and electrons are passing the anode with a speed of $v= \frac{1}{2}\cdot c$. Find the acceleration voltage.
  4. Concerning the non-relativistic calculation, which acceleration voltage is needed to accelerate electrons to $v= \frac{1}{2}\cdot c$ ? What is the relative error of this calculation?